So good morning, good morning.

We are colleagues and friends, friends and colleagues.

Welcome to Erlangen, welcome to Frederik Suleisander Universitat Erlangen, Nuremberg.

As you know, we are here on Erlangen campus, we have another campus in Nuremberg, also

other campuses in the area.

This is why the name of the university is Frederik Suleisander Universitat on the memory

of the founder of the university, plus Erlangen, Nuremberg.

So Emy Nother was born here in Erlangen.

You can actually visit the birth home of Emy Nother, Emy Nother mathematician that we all

know.

She used to say that the goal of mathematicians should not be to study objects, but rather

connections between objects.

And probably this is what inspired her celebrated theorem on the connection between symmetries

and conservation laws and invariances.

So after she got the PhD degree here at this university, she moved to Gottingen.

In Gottingen, he was in particular collaborating with David Hilbert and Felix Klein.

So it was a dream team.

Actually Felix Klein had been a professor here in Erlangen before, and this is why he

wrote his celebrated Erlangen program.

As you know, Felix Klein was very broad, he was not only interested in mathematical research,

but also in the history, education, he had this holistic view about mathematics.

He was working in this hyperbolic geometry, and he wrote this paper on this surface.

We were seeing the surface now, of course, according to the Murphy's principle, is gone

whenever I wanted to point to it.

But you have seen this bottle, it's kind of a three-dimensional Moebius band, so it's

a surface without an entrance or an exit.

As a bottle will not be a very interesting object, you cannot feel it, and if you can

feel it, then you cannot drink out of it.

But his paper was on this Flache, so Flache of Deutsch in Germany's surface.

But it seems that the translator, I mean, it was handwritten, the translator, when reading

this word Flache, and maybe influenced by the fact that he was visiting Erlangen and

he observed that beer was here very popular and very good, he understood it was written

Flache, so there is an extra S on the German word for bottle.

And now, because the surface looked like a bottle, then it remained like Klein bottle

that we know.

So therefore, connecting fields was one of the goals of all these individuals.

Of course, we still live on the ideas of David Hilbert, so if David Hilbert didn't say that

the Laplacian can be solved by minimizing an energy, probably we will not be here, right?

I mean, all we have learned starts by the fact that the Laplace equation can be solved

in Sobolev spaces by minimizing an energy, right?

But of course, this needed a lot of effort for generations, right?

So building Sobolev spaces, the theory of distributions, actually things are very curious,

right?

Because Sobolev was probably before Loran Svarts, who was the Fields Medal for the Theory of

Distributions, so well, it took some time until people figured out how things could

be done.

Now with machine learning, you don't need all this, right?

To some extent.

And this is why we are here in the right place to build connections, right?

So following this spirit of Felix Klein and Amy Noether, between different fields, and